 Solution

The first-order multivariable Taylor expansion isTaking $(x_0,y_0)=(2,0)$ gives the linear approximationTo assess its significant figures, the quadratic Taylor polynomial supplies the correction $h^2+4hk+2k^2=-0.0001$ for $h=0.03$, $k=-0.01$. Direct evaluation gives $2.03^2{e}^{-0.01}=4.0798963599\dots$. Thus the estimate is correct to four significant figures: both round to $4.080$, whereas to five significant figures the exact value is $4.0799$ and the approximation is $4.0800$. The unexpectedly small quadratic correction results partly from cancellation of its mixed and unmixed terms.